Page 13 - Mathematical Techniques of Fractional Order Systems

Variable Order Fractional Derivatives and Bone Remodeling Chapter | 1 3

1.1.2 Application Example of Variable Order Derivatives: Bone
Remodeling

Bone is constantly being renewed, being destroyed, and formed, due to cells
termed osteoclasts and osteoblasts, respectively. In the adult skeleton, both
processes are in balance and tightly coupled through autocrine and paracrine
factors between bone cells that allow for a constant bone density to be main-
tained. As this specific microenvironment provides the necessary conditions
for the growth and proliferation of tumor cells, bone is a common site for
the development of metastases, mainly from primary breast and prostate
cancer.
Mathematical and computational models, with differential equations that
represent the control mechanisms involved, can replicate this remodeling
process (Komarova et al., 2003). These models have been extended to
include the effects of tumor disruptive pathologies in the bone dynamics, as
metastases contribute to the decoupling between bone resorption and forma-
tion and to the self-perpetuating tumor growth cycle (Ayati et al., 2010).
Counteraction effects of currently used therapies were also contemplated
(Ayati et al., 2010), through the pharmocokinetic (PK) and pharmacody-
namic (PD) combination of anticancer (chemotherapy) with antiresorptive
treatments (bisphosphonates or monoclonal antibodies) (Christ et al., 2018;
Coelho et al., 2016, 2015).
Fractional and variable order derivatives can be successfully used in
modeling the dynamics of bone remodeling. Christ et al. (2018) implemented
fractional derivatives in the differential equations of bone remodeling pre-
sented in Ayati et al. (2010), and analyzed its dynamic behavior in the
absence and presence of tumor and PK/PD applied treatment, for a discre-
tized single point and a one-dimensional bone. Vale ´rio et al. (2016) applied
the same methodology to the more accurate biochemical model of Coelho
et al. (2016). Variable order derivatives have also been introduced in Neto
et al. (2017), as a simplification technique, in the models of Ayati et al.
(2010) and Coelho et al. (2015), in an effort to replicate the same bone
microenvironment response but recurring to less parameters to impose the
known bone behavior. It is the latter that is here revisited and further
analyzed.

1.1.3 Chapter Organization

The remaining sections of this chapter are organized as follows. Variable
order concepts and definitions are addressed in Section 1.2. Bone remodeling
physiology, PK/PD concepts, and published integer mathematical models are
presented in Section 1.3. Variable order bone models are address in
Section 1.4. Finally, conclusions are presented in Section 1.5.